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Mirrors > Home > MPE Home > Th. List > fmptapd | Structured version Visualization version GIF version |
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.) |
Ref | Expression |
---|---|
fmptapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fmptapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fmptapd.s | ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) |
fmptapd.c | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
fmptapd | ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptapd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) | |
2 | fmptapd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | fmptapd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 1, 2, 3 | fmptsnd 7189 | . . 3 ⊢ (𝜑 → {〈𝐴, 𝐵〉} = (𝑥 ∈ {𝐴} ↦ 𝐶)) |
5 | 4 | uneq2d 4178 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
6 | mptun 6715 | . . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
8 | fmptapd.s | . . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) | |
9 | 8 | mpteq1d 5243 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
10 | 5, 7, 9 | 3eqtr2d 2781 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {csn 4631 〈cop 4637 ↦ cmpt 5231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-mpt 5232 |
This theorem is referenced by: fmptpr 7192 poimirlem3 37610 poimirlem4 37611 poimirlem16 37623 poimirlem17 37624 poimirlem19 37626 poimirlem20 37627 |
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